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0v2

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Jun

2001

hep-th/0103080HUTP-01/A012

CTP-MIT-3094

CTP TAMU-09/01

Supergravity duals of gauge theories from F (4)gauged supergravity in six dimensions

Carlos Nuneza, I.Y. Parkb†, Martin Schvellingerc, Tuan A. Tranb‡

aLyman Laboratory of Physics,

Harvard University,

Cambridge, Massachusetts 02138, USA

E-mail: nunez@lorentz.harvard.edu

bCenter for Theoretical Physics,

Texas A&M University

College Station, Texas 77843, USA

†E-mail: ipark@rainbow.physics.tamu.edu

‡E-mail: tuan@rainbow.physics.tamu.edu

cCenter for Theoretical Physics,

Laboratory for Nuclear Science and Department of Physics,

Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

E-mail: martin@ctpbeaker.mit.edu

Abstract

We construct supergravity solutions dual to twisted field theories that are the world-

volume theories of D4-branes wrapped on 2, 3-cycles, and NS-fivebranes on 2-cycles.

We first obtain the solutions for the Romans’ six-dimensional gauged supergravity the-

ories and then up-lift them to ten dimensions. In particular, we find solutions for field

configurations with either non-Abelian fields or B-fields being excited. One of these

solutions, in the massless case, is up-lifted to the massless type IIA string theory. This

is the first example of such a kind. The cases studied provide new examples of the

AdS/CFT duality involving twisted field theories.

1 Introduction

The large N -limit of superconformal field theories have dual descriptions in terms of

weakly coupled supergravities [1, 2, 3]. Generalizations to non-conformal cases were

also studied [4]. There is an extensive list of references regarding the flows of N = 4

super Yang-Mills theory under the perturbations of either masses or vev’s for bilinear

operators [5, 6]. With the twisted gauge field theories [7] available, it is possible to

find new examples of AdS/CFT where only a part of the maximal supersymmetries

is preserved [8, 9, 10, 11, 12, 13]. This allows us to investigate supergravity duals of

partially twisted gauge field theories. Therefore, wrapped fivebranes and D3-branes on

holomorphic curves were studied in [8, 9]. Similarly, fivebranes [10] and D3-branes [12]

wrapped on associative three-cycles were considered. These studies have been extended

to M-fivebranes wrapping Kahler 4-cycles, special Lagrangian 3, 4 and 5-cycles, co-

associative 4-cycles and Cayley 4-cycles [11]. In this paper, we concentrate on the D4-

D8 system, which was studied in depth from the gauge theory side in ref. [14] and from

the gravity/CFT viewpoint in refs. [15, 16]. We will wrap D4-branes of a D4-D8 system

on two and three-cycles. We will also study solutions corresponding to smeared NS-

fivebranes wrapped on two-cycles. We first find solutions of six-dimensional Romans’

gauged supergravity [17] and up-lift them to massive type IIA string theory. Other

constructions in massive type IIA supergravity can be found in ref. [18], where some

intersecting-brane configurations of D0-branes, D8-branes and a string were studied.

Then, we analyze their dual twisted field theories and study the flows from AdS6-type

regions to the AdS6−p × Σp solutions, where Σp is the p-dimensional curved manifold.

In the present case we consider p = 2 and 3.

The Romans’ theories we are interested in are the SU(2) gauged N = 4 supergravi-

ties in six dimensions. There are four different theories depending upon the values of

the SU(2) coupling constant g and the mass parameter m of the two-form field B. One

of these theories has two anti-de Sitter vacua, one of them breaking all the supersym-

metry and the other one preserving all of it. The latter theory, with a ground state

with unbroken F (4) supersymmetry, is the six-dimensional supergravity predicted in

[19, 20]. In this paper, we will consider this case.

We start out to find solutions of six-dimensional gauged supergravity with vanishing

B-fields, using a hyperbolic metric ansatz of the type AdS4 ×H2 (and also a spherical

one, for AdS4 × S2). We find a fixed-point solution in the massive case, whereas for

massless field configurations we find a general solution. The latter has “bad” type

singularities: in order to resolve the singularities we look for solutions with additional

degrees of freedom in a similar way to ref. [9]. This leads us to look for non-Abelian

solutions. Another possible way to resolve singularities is to find a black hole solution

where the singularity is covered by a horizon [21]. The aim of this paper is to find the

1

above mentioned gravity solutions, perform the up-lifting of them to ten dimensions

and, to study their dual (twisted) gauge field theories. Massive type IIA supergravity

is an interesting theory which seems to be the most suitable arena for realizing de

Sitter compactifications or Randall-Sundrum models [22] on manifolds with boundary.

Therefore, the study of the systems like the ones we deal with here could give some

ideas on general structures.

Since in order to study the flow between a higher dimensional theory and the re-

sulting lower dimensional one it is necessary to compactify some of the directions on a

curved manifold, spinors have to be defined on non-flat spaces. Therefore, this requires

to deal with a twisted gauge field theory living in such a manifold. Thus, in order

to consider the dual field theories corresponding to certain supergravity solutions in

curved spaces, one has to study a twisted version of the corresponding supersymmetric

theory. This new picture emerges from the impossibility in defining covariantly con-

stant spinors in the aforementioned manifolds. Then, it is necessary to introduce a

new term coming from the gauging of the normal bundle to the brane world-volume

with the spin connection.

We consider spacetimes with geometries given by AdS4 × Σ2 and AdS3 × Σ3, such

that their UV limit is an AdS6-type spacetime. These lead to twisted field theories

since we assume Σp’s are curved manifolds. The decoupling limit is taken in such a way

that it preserves the volume of the compact manifold. Then, the twisted field theory

on the brane world-volume will be R1,4−p × Σp, for D4-branes wrapping Σ2 and Σ3.

The field theory in this decoupling limit will not be sensitive to the global geometry of

Σp, remaining its effect local and determining the twisted field theory [8]. For energies

much lower than the radius of the compact manifold the theory will be a 3 (or 2)

dimensional field theory, respectively.

This paper is organized as follows. In section 2 we review the basic formalism and

set-up of the six-dimensional Romans’ theories. Section 3 is devoted to the study of

the gravity duals of superconformal field theories in 3 and 5 dimensions. We focus on a

non-Abelian solution, which will be up-lifted to massive type IIA string theory. It can

be interpreted as a D4-D8 system where the D4-brane is wrapped on 2- and 3-cycles.

Section 4 begins with the up-lifting for the massless case to type IIA theory (even when

expected this is another new result of this work). Then, we study the corresponding

twisted gauge field theories. We discuss the relation between our solutions and the

previous ones [10], regarding the structure of their actions in the string frame. Finally,

we present our conclusions.

2

2 The Romans’ theories in 6 dimensions

In this section we review the six-dimensional gauged N = 4 supergravity constructed

by Romans [17], whose conventions we follow. The theory consists of a graviton eαµ,

three SU(2) gauge potentials AIµ, an Abelian potential Aµ, a two-index tensor gauge

field Bµν , a scalar φ, four gravitinos ψµ i and four gauginos χµ. The bosonic Lagrangian

is

e−1 L = −1

4R +

1

2(∂µφ) (∂µφ) − 1

4e−

√2φ (Hµν Hµν + F I µν F I

µν)

+1

12e2

√2φGµνρG

µνρ +1

8(g2 e

√2φ + 4g m e−

√2φ −m2 e−3

√2φ)

−1

8e εµνρστκ Bµν (Fρσ Fτκ +mBρσ Fτκ +

1

3m2Bρσ Bτκ + F I

ρσ FIτκ), (1)

where e is the determinant of the vielbein, g is the SU(2) coupling constant, m is the

mass parameter associated with the two-index tensor field Bµν and εµνρστκ is a Levi-

Civita tensor density. The Abelian field strength Fµν , the non-Abelian one F Iµν , the

three-form Gµνρ and the field Hµν are given by

Fµν ≡ ∂µAν − ∂νAµ ,

F Iµν ≡ ∂µA

Iν − ∂νA

Iµ + g ǫIJK AJ

µ AKν ,

Gµνρ = 3 ∂[µBνρ] ,

Hµν ≡ Fµν +mBµν , (2)

respectively. The supersymmetry transformations for the gauginos and gravitinos are

δχi =

(

1√2γµ ∂µφ+ Aγ7 −

1

12e√

2φγ7 γµνρGµνρ

)

ǫi +1

2√

2γµν (Hµν)

ji ǫj , (3)

δψµ i =(

∇µ + T γµ γ7 −1

24e√

2φ γ7 γνρσ Gνρσ γµ

)

ǫi

+

(

g AIµ (T I) j

i − 1

4√

2(γ νρ

µ − 6 δ νµ γρ)(Hνρ)

ji

)

ǫj , (4)

where A, T , and H are defined as follows

A ≡ 1

4√

2(g e

φ√

2 − 3m e−3φ√

2 ), T ≡ − 1

8√

2(g e

φ√

2 +m e−3φ√

2 ) , (5)

(Hµν)j

i ≡ e− φ

√

2

(

1

2Hµν δ

ji + γ7 F

Iµν (T I) j

i

)

. (6)

The gauge-covariant derivative Dµ acting on the Killing spinor is

Dµ ǫi = ∇µ ǫi + g AIµ (T I) j

i ǫj , (7)

3

with

∇µǫi ≡ (∂µ +1

4ω αβ

µ γαβ) ǫi , (8)

where ω αβµ is the spin connection. Indices α, β are tangent space (or flat) indices,

while µ, ν are spacetime (or curved) indices. The γαβ··· are the six-dimensional Dirac

matrices,

γα1···αn=

1

n !γ[α1

· · · γαn], n = 1, · · · , 6.

The equations of motion of the Lagrangian (1) are

Rµν = 2 ∂µφ ∂νφ+1

8gµν (g2 e

√2φ + 4 gm e−

√2φ −m2 e−3

√2φ)

+e2√

2φ(G ρσµ Gνρσ − 1

6gµν G

ρστ Gρστ )

−2 e−√

2φ(H ρµ Hνρ −

1

8gµν Hρσ Hρσ)

−2 e−√

2φ(F I ρµ F I

νρ −1

8gµν F

Iρσ F

I ρσ), (9)

2φ =1

4√

2(g2 e

√2φ − 4mg e−

√2φ + 3m2 e−3

√2φ) +

1

3√

2e2

√2φ

×Gµνρ Gµνρ +1

2√

2e−

√2φ (Hµν Hµν + F I µν F I

µν), (10)

Dν (e−√

2φ Hνµ) =1

6e εµνρστκ HνρGστκ, (11)

Dν (e−√

2φ F I νµ) =1

6e εµνρστκ F I

νρGστκ, (12)

Dρ (e2√

2φGρµν) = −m e−√

2φ Hµν − 1

4e εµνρστκ (Hρσ Hτκ + F I

ρσ FIτκ). (13)

Depending upon the values of the gauge coupling and mass parameter, there are five

distinct theories: N = 4+ (for g > 0, m > 0), N = 4− (for g < 0, m > 0), N = 4g

(for g > 0, m = 0), N = 4m (for g = 0, m > 0), and N = 40 (for g = 0, m = 0).

The N = 4g theory coincides with a theory [23] obtained by dimensional reduction of

gauged N = 2 supergravity in seven dimensions, followed by truncation. It was pointed

out in [17] that there is a dual version of the N = 4g theory, which has a similar field

content but cannot be obtained from the N = 4g theory by a field redefinition. The

Bµν field is replaced by a new tensor field Aµν , whose field strength is F Iµν . In absence

of the Abelian field strength, the bosonic Lagrangian of the dual N = 4g theory is

e−1 L = −1

4R+

1

2∂µφ∂

µφ− 1

4e−

√2φ F I

µνFI µν +

1

12e−2

√2φ FµνρF

µνρ +1

8g2 e

√2φ, (14)

where

Fµνρ ≡ 3(

∂[µ Aνρ] − F I[µν A

Iρ] −

1

3g ǫIJK AI

µ AJν A

Kρ

)

. (15)

4

The Lagrangian (14) can be obtained from the Lagrangian (1) by formally writing Fµνρ

as follows

Fµνρ =1

6e2

√2 φ e εµνρστκG

στκ . (16)

Since the Lagrangian (14) differs from N = 4g theory only by a sign of dilaton coupling

to three-form field, in the absence of the three-form field N = 4g and N = 4g are

identical. Furthermore, the Eq. (16) tells us that any solution of N = 4g theory

with the two-form fields not being excited, which is also a solution of N = 4g theory,

can be up-lifted into higher dimensional theory. We will consider the ten-dimensional

interpretation of N = 4g theory in section 4.

3 Duals of 5 and 3-dimensional SCFTs

In this section we study the gravity duals of superconformal field theories in five dimen-

sions with 8 supercharges [14], which in the IR flow to superconformal field theories in

three dimensions with 4 supercharges, which is N = 2 in three dimensions [24, 25].

The spinors of SO(1, 4) are pseudoreal with four components. These theories in 5

dimensions are related by compactifications to N = 2 four-dimensional theories. They

have an SU(2)R symmetry of automorphism algebra which can be associated with

the gauged symmetry in the gravitational set-up of [17]. The vector multiplet in five

dimensions has one real scalar component, a vector gauge field and a spinor, while the

hypermultiplet has four real scalars and a fermion. This theory has a Coulomb branch

when the real scalar has a vev, and a Higgs branch when the scalar in the hypermultiplet

is excited. Our gravitational system can be understood as follows. Suppose that we

start from type I theory on R9 × S1, and we consider N D5-branes, the gauge theory

on the branes is N = 1 Sp(N) gauge field theory with one hypermultiplet in the

antisymmetric representation and 16 hypermultiplets in the fundamental. After T-

dualizing on S1 we arrive at a type I’ configuration on S1/Z2 with two O8 planes, N

D4-branes and 16 D8-branes. The hypermultiplet in the antisymmetric representation

is massless while the mass of the fundamental hypermultiplets is given by the relative

position of the D8-branes with respect to the D4-brane. The theory on the D4-brane

is Sp(N) with one vector multiplet (whose scalar component describes the Coulomb

branch R+) and hypermultiplets whose first components describe the Higgs branch.

The global symmetries of the theory are SU(2)R×SU(2)×SO(2Nf)×U(1), being the

first SU(2) the R-symmetry (the supercharges and the scalars in the hypermultiplets

are doublets under this group). The second one is associated with the hypermultiplet

in the antisymmetric representation and the rest is associated with the hypermultiplets

in the fundamental and instantons. When the D4-brane is in the origin of the Coulomb

branch we have a fixed point and the global symmetry is enhanced to SU(2)×ENf +1.

5

The gravitational system is given by Nf D8-branes situated on a O8 plane, 16 − Nf

D8-branes in another fixed plane and, a D4-brane which can move between them. The

position of the D4-brane is parameterized by the scalar in the vector multiplet. If

< φ > is not zero we have a theory with a U(1) symmetry with Nf “electrons”, on the

fixed planes, the theory recovers its SU(2) R-symmetry and it will have Nf quarks.

The gravity theory is a fibration of AdS6 over S4, with isometries SO(2, 5)×SU(2)×SU(2). The six-dimensional SU(2) gauged supergravity has an AdS6 vacuum solution.

It can be up-lifted to massive IIA theory [26] leading to the configuration described

above [12]. In our case we will consider solutions of Romans’ theory which, when

uplifted, will give ten dimensional configurations with Nf = 0. These theories were

very well studied in [30] in the context of algebraic geometry. There the authors

studied the moduli spaces of type I’ string and Heterotic string. It will be interesting

to understand the case of Nf 6= 0 and see whether there is a gravity solution dual to

the theory with enhanced symmetry E ′1. It will be also useful to clarify the role of

D0-branes in our gravity solutions. In principle they should correspond to excitations

of the six-dimensional Abelian gauge field. We will not consider these fields here. We

leave this for a future work.

In order to analyze the flow of the theory from five dimensions to a superconformal

theory in three dimensions with four supercharges, we would like to consider a configu-

ration where the geometry “loses” two dimensions at low energies. Therefore, the idea

is the following: we start with a six-dimensional gauged supergravity theory having an

AdS6 vacuum and 8 preserved supercharges. This vacuum solution is dual to the five-

dimensional SCFT mentioned above and its ten-dimensional interpretation is given by

a D4-D8 system. When one moves to the IR of the gauge theory (flowing in the radial

coordinate on the gravity dual) two of the dimensions of the theory become very small

and no low energy massless modes are excited on this two-space. Therefore, effectively

the gauge theory is three-dimensional. Besides, since the D4-brane is wrapped on a

curved surface, we need to twist the theory. The effect of this twisting is the breaking

of some supersymmetries, as it can be seen from the projections of the Killing vectors.

Under the group SO(1, 4) × SO(3)R the fields on the D4-brane transform as (1, 1),

(3, 1) and (4, 2). In the IR the metric is taken to have SO(1, 2)× SO(2)D symmetry.

The twisting involve “mixing” the SO(2) group of the metric with an SO(2) included in

SO(3)R. This mixing is responsible for the breaking of some supersymmetries because

the only spinors that can be defined on the curved manifold are those that have scalar

properties on the curved part. In fact, it breaks 1/2 of the supersymmetries of the

five-dimensional SCFT.

Below we construct a gravity dual to the five-dimensional SCFT theory by finding a

solution of Romans’ theory with excited F Iµν fields. Then, we find a fixed-point solution

for the massive field configuration. We postpone the study of the massless case to the

6

next section.

Let us consider the metric ansatze of the form

ds2 = e2 f (dt2 − dr2 − dz2 − dv2) − e2 h

y2(dx2 + dy2) , (17)

ds2 = e2 f (dt2 − dr2 − dz2 − dv2) − e2 h (dθ2 + sin2 θ dϕ2) , (18)

for AdS4×H2 and AdS4×S2, respectively. The only non-vanishing non-Abelian gauge

potential components are taken to be

A(3)x =

a

y, A(3)

ϕ = −a cos θ , (19)

in the hyperbolic and spherical cases, respectively. One can treat both cases together

by introducing a parameter λ that can take two values

λ = +1 → H2, λ = −1 → S2 . (20)

We impose the following projections

γ45(T(3)) j

i ǫj =λ

2ǫi, γ2γ7ǫi = ǫi . (21)

A solution must satisfy the equations obtained by setting to zero the supersymmetry

transformations for gauginos and gravitinos. After some algebra, they lead to

ϕ′ =ef

4√

2[g eϕ − 3m e−3ϕ + 2 a λe−2h−ϕ] , (22)

h′ =ef

4√

2[−g eϕ −m e−3ϕ + 6 a λ e−2h−ϕ] , (23)

and

f ′ = − ef

4√

2[g eϕ +m e−3ϕ + 2 a λ e−2h−ϕ] , (24)

where ϕ ≡ φ/√

2.

A fixed-point solution of these equations is given by

e4ϕ =2m

g,

e−2h =

√

g m

8

1

aλ,

e−f(r) =g eϕ

2√

2r . (25)

7

In fact, it only happens if λ a > 0. This solution satisfies the second order equations

derived from the starting six-dimensional Lagrangian.

We can analyze the structure of the solutions of the system above near the UV of the

theory (r = 0). Indeed, we can see that an expansion leads to the following behavior

for the different fields

f ≈ − log(r) + c1r2 + ..., g ≈ − log(r) + c2r

2 + ..., φ ≈ c3r2 + ... (26)

(where ci’s are constants). The interpretation of these expansions [27] is that the

scalar field φ describes the coupling of the five-dimensional field theory to an operator

of conformal weight ∆ = 3 and mass squared m2 = −6. The operator is not the

highest component of its supermultiplet, thus it is a deformation that breaks some

supersymmetry.

The ten-dimensional metric corresponding to the fixed-point solution described above

is easily obtained by using the results of [26],

X =

(

2m

g

)1/4

, ∆ = g X cos2 χ+ 3mX−3 sin2 χ , (27)

ds210 = (3mg2)−1/8 (sinχ)

1

12X1/8

[

∆3/8 e2f0

2 r2(−dt2 + dr2 + dv2 + dz2)

+

√2∆3/8

√g3my2

(dx2 + dy2) + ∆3/8√

2mg3 dχ2

+∆−5/8

g Xcos2 χ

(

σ1 −dx

y

)2

+ σ22 + σ2

3

. (28)

The metric has symmetries of the form SO(2)×SO(3). The corresponding expression

for the RR fields and the ten-dimensional dilaton can be read from [26] and we do not

quote it here.

Although we could not find an exact solution, we can carry out a similar analysis

to [10]. Actually, the two cases λ = ±1 can be gathered again. Thus defining F ≡u2e−2ϕ , u ≡ e2h(r) and using Eqs.(22), (23) we obtain the following differential equation

dF

du=

(

3gu4 −mF 2 − 10aλuF

gu4 +mF 2 − 6aλuF

)

F

u. (29)

For the case λ = +1 (hyperbolic plane), we can solve this equation in the approxima-

tions when u→ ∞ and u→ 0. In fact, in the UV one has

F ≈ u2 + ..., e−2ϕ ≈ 1 + ..., (30)

8

and the metric results to be the AdS6 limit of our original metric. In the IR we have

F ≈ F0u5/3 + ..., e−2ϕ ≈ F0u

−1/3 + ... (31)

and the metric is

ds26 =

1

u1/3(dt2 − dz2 − dv2) − u

y2(dx2 + dy2) − 2

9F0 a u1/3du2 . (32)

0.2 0.4 0.6 0.8 1u

0.2

0.4

0.6

0.8

1F

UVAdS6IR

BS

Figure 1: Orbits for the case AdS4 × H2. In the UV limit F behaves as u2 leading to anAdS6-type region. On the other hand, it flows to “bad” singularities (BS) in the IR. We haveused g = 3m with m =

√2, and a = 1.

When up-lifted the singularities turn out to be of the “bad”type. They correspond

to the curves flowing to the origin of the plots in figure 1. On the other hand, in both

cases of the hyperbolic plane and sphere there are other kinds of “bad” singularities

that correspond to the orbits going like F ≈ 1u

as u approaches to zero. Particularly,

one can see this kind of behavior in figure 2. For the two-sphere we have the set of

orbits depicted in figure 2.

This also constitutes a part of the IR behavior of the flow. As we can see in figures

1 and 2, there is a smooth solution interpolating between the UV and the IR limits

of the system, thus realizing our initial set up. The fixed-point solution is out of the

range of this plot.

9

0.2 0.4 0.6 0.8 1u

0.2

0.4

0.6

0.8

1F

UVAdS6

IRBS

Figure 2: We show the behavior of the orbits for the case S2. Similar comments as in theprevious case are also valid here. For small u-values we can see the region corresponding toF ≈ 1

u .

In summary, we have found a fixed-point solution for the hyperbolic plane. In the

UV both geometries flow to their corresponding asymptotic AdS6. In the IR limit all

the singularities in both geometries are of “bad” type according the criterion of ref.

[8]. Similar behaviour is expected for the sphere. Note that the solutions studied here

only represent a special case of solutions that, in general, have the form of a Laurent

series. The fact that these solutions have non-acceptable singularities can be easily

seen from the construction of an effective potential corresponding to this effective four

dimensional system, and following the analysis outlined by Gubser in ref. [28] one can

see that these kinds of solutions will not be interpretable as the IR of a gauge theory.

However, this does not exclude the possibility of finding a different solution that could

have an interpretation as the transcendental solution found in ref. [10].

3.1 A non-Abelian solution in massive type IIA theory

In this section we describe a non-Abelian solution in the massive IIA theory. There-

fore, we consider a gravity dual of a five-dimensional SCFT with 8 supercharges that

flows through the dimensions to a two-dimensional (1,1) CFT with two supercharges

at the IR. Its field content is given by a real scalar, a gauge field and the correspond-

ing fermionic partner. The idea is to find a compactification from six-dimensional

supergravity on AdS3 ×H3. Under SO(1, 4)× SO(3)R the charges transform as (4, 2)

and (4, 2). After the twisting SO(1, 4) × SO(3)R → SO(1, 1) × SO(3) × SO(3)R →SO(1, 1) × SO(3)D we are left with two supercharges which are scalars under the di-

agonal subgroup. SO(3)D results from the identification of the normal bundle of the

D4-brane world-volume with the spin bundle of Σ3. They are the two “twisted” su-

10

percharges that survive the twisting process. The field content of the theory can be

obtained along the same lines and the result is the one mentioned above.

We believe that this solution is the first example of a non-Abelian solution of mas-

sive type IIA theory. It is similar to the solution obtained in [10] for massive seven-

dimensional gauged supergravity. In that case the solution up-lifts to M theory. There-

fore, this suggests some connection between theories in the line of ref. [31] that merits

further exploration.

Our configuration is

ds2 = e2f (dt2 − dr2 − du2) − e2h

y2(dx2 + dz2 + dy2), A(1)

x =a

y, A(3)

z =b

y, (33)

we can also consider an AdS3 × S3, for λ = −1. The projections (all the indexes are

plane indexes) are given by

γ65(T(3))j

i ǫj =1

2ǫi, γ64(T

(2))ji ǫj =

1

2ǫi, γ45(T

(1))ji ǫj =

1

2ǫi, γ7γ2ǫi = ǫi , (34)

we use ξ = eφ√

2 . The BPS equations are

a b =1

g, (35)

h′ = − ef

4√

2[gξ +mξ−3 − 10λ aξ−1 e−2h] , (36)

φ′ =ef

4√

2[gξ − 3mξ−3 + 6λ a ξ−1 e−2h] , (37)

f ′ = − ef

4√

2[gξ +mξ−3 + 6λ a ξ−1 e−2h] . (38)

A fixed-point solution is obtained and it reads as it follows

e−2h =

√g3m

2√

6, ξ =

(

3m

2g

)1/4

. (39)

This configuration can be uplifted to ten dimensions (massive IIA theory) using the

results in [26]

X = eφ/√

2,∆ = g X cos2 χ+ 3mX−3 sin2 χ , (40)

ds210 =

(

1

3mg2

)1/8

(sinχ)1

12X1/8

[

∆3/8

2 r2(−dt2 + dr2 + dv2)

+e2h

2 y2∆3/8 (dx2 + dz2 + dy2) + g2 ξ∆3/8 dχ2

+∆−5/8

g Xcos2 χ

(

σ1 −dx

y

)2

+ σ22 +

(

σ3 −dz

y

)2

, (41)

11

where σi are the three left-invariant forms in the three-sphere. We can see that the

metric has an SO(2) × SO(3) invariance.

Since we are not able to integrate the above BPS system, we will repeat the analysis

of the singularities as we did in the previous section. Again, both cases (hyperbolic

plane and sphere) are treated together by using the parameter λ = ±1. Proceeding as

above we get the following first order differential equation

dF

du=

(

3gu4 −mF 2 − 14aλuF

gu4 +mF 2 − 10aλuF

)

F

u. (42)

When λ = +1 we solve this equation in two approximations, u → ∞ and u → 0. The

large-u approximation leads to the corresponding asymptotic AdS6 whereas the other

case gives

F ≈ F01

u+ ..., e−2ϕ ≈ F0

u3+ ..., (43)

F ≈ −2a

mu+ ..., e−2ϕ ≈ − 2a

mu+ ... (44)

F ≈ F0u7/5 + ..., e−2ϕ ≈ F0u

3/5 + ... . (45)

Note that all the singularities are of the “bad” type in the IR.

0.2 0.4 0.6 0.8 1u

0.2

0.4

0.6

0.8

1F

\ 6 \ \ \ \ \

x

Figure 3: Behavior of the orbits for the non-Abelian massive case. The AdS6-type region isat the UV, i.e., for large u and F -values. In the IR it flows to “bad” singularities (BS). Thecross indicates the fixed point-solution aforementioned. We again set g = 3m, m =

√2, and

a = b = λ = 1.

Here we should make a similar comment as the one in the previous section: Gubser’s

criteria tells us to construct the effective potential of the three-dimensional gravity

12

theory. With this potential we can see that the solutions in the form of a Laurent

series showed above will not have an interpretation as the IR limit of a gauge theory.

4 The massless case

Consider six-dimensional gauged supergravity with the mass parameter taken to be

zero (m = 0). We would like to find a ten-dimensional Kaluza-Klein ansatz which

allows us to perform the up-lifting of certain solutions of the six-dimensional Romans’

theory. In this section, we will show that the N = 4g theory can be embedded in type

IIA theory.

Thus, we start with type IIA theory in ten dimensions and set the two- and three-

form field strengths to zero. After the truncation the bosonic Lagrangian is

L10 = R ∗ 1 − 1

2∗ dφ ∧ dφ− 1

2e−

1

2φ ∗ F(4) ∧ F(4) . (46)

The ten-dimensional equations of motion are

d∗dφ = −1

4e−

1

2φ ∗ F(4) ∧ F(4),

d(e−1

2φ∗F(4)) = 0,

Rµν =1

2∂µφ∂ν φ+

1

12e−

1

2φ (Fµρστ F

ρστν − 3

32gµν F

2(4)),

F(4) ∧ F(4) = 0 . (47)

Notice that the last equation is a constraint resulting from the fact that we set F(3) =

F(2) = 0.

Reducing the theory on a circle, we get a nine-dimensional theory whose bosonic

sector consists of a graviton, two scalars, a three-form field strength and a two-form

field strength. One can consistently truncate this theory to a theory whose bosonic

content includes a graviton, a scalar and a three-form field strength. The ansatze for

the metric, scalar and three-form fields are

ds210 = e

− 5

8√

14φds2

9 + e35

√

14φdZ2,

φ =

√14

8φ,

F(4) = F(3) ∧ dZ . (48)

The resulting nine-dimensional Lagrangian and its equations of motion are

L = R ∗ 1 − 1

2∗ dφ ∧ dφ− 1

2e− 4

√

14φ ∗ F(3) ∧ F(3),

13

0 = d(e− 4

√

14φ∗F(3)),

2φ = − 1

3√

14e− 4

√

14φF 2

(3),

Rµν =1

2∂µφ ∂ν φ+

1

4e− 4

√

14φ(

FµρσFρσ

ν − 2

21gµνF

2(3)

)

. (49)

Now, we compactify the 9-dimensional theory on S3 using the formulae in [29]. The

metric ansatz is

ds29 = X3/14ds2

6 +1

4 g2X−2/7

3∑

i=1

(σi − g Ai(1))

2, (50)

where σi, i = 1, 2, 3 are left-invariant 1-forms of SU(2)

σ1 = cosψ dθ + sinψ sin θ dϕ,

σ2 = − sinψ dθ + cosψ sin θ dϕ,

σ3 = dψ + cos θ dϕ, (51)

and X = exp(√

2 φ). The ansatze for three-form field, its Hodge dual and the scalar

field are

e√

7

2φ = X ,

F(3) = F(3) −1

4 g2h1 ∧ h2 ∧ h3 +

1

4

3∑

i=1

F i(2) ∧ hi,

e− 4

√

14φ ∗ F(3) = − 1

8g3X−1 ∗ F(3) ∧ h1 ∧ h2 ∧ h3 + 2 g X1/2 ǫ(6)

+1

16g2ǫijk X

−1/2 ∗ F i(2) ∧ hj ∧ hk, (52)

where hi ≡ σi − g Ai. Substituting these ansatze into the nine-dimensional equations

of motion we get

dF(3) =1

2

3∑

i=1

F i(2) ∧ F i

(2),

d(X−1 ∗ F(3)) = 0, (53)

D(X−1/2 ∗ F i(2)) = X−1 ∗ F(3) ∧ F i

(2),

d(X−1 ∗ dX) = X−1 ∗ F(3) ∧ F(3) +1

4X−1/2

3∑

i=1

∗F i(2) ∧ F i

(2) − 4g2X1/2 ǫ(6).

The ansatz and the Lagrangian that produce Eq. (54) are

ds210 = e

√

2

8φds2

6 +1

4g2e−

3√

2

8φ

3∑

i=1

(σi − g Ai(1))

2 + e5√

2

8φdZ2,

14

F(4) = (F(3) −1

4g2h1 ∧ h2 ∧ h3 +

1

4gF i

(2) ∧ hi) ∧ dZ,

φ =1

2√

2φ,

1√gL6 = R− 1

2(∂φ)2 − 1

4e− 1

√

2φF i

µνFi µν + 4 g2e

1√

2φ − 1

12e−

√2φF 2

(3) , (54)

where F(3) = dA(2) + 14(F i

(2) ∧ Ai(1) − 1

6g ǫijk A

i(1) ∧ Aj

(1) ∧ Ak(1)). For an easy comparison

with the Romans’ theory we make the following rescalings

gµν → −1

2gµν , g →

1

2g, Ai →

√2Ai, φ→ 2φ, F3 → F3 . (55)

The Lagrangian and the ansatz in Eq.(54) now become

ds210 = −1

2e

√

2

4φ ds2

6 +1

g2e−

3√

2

4φ

3∑

i=1

(

σi − 1√2g Ai

(1)

)2

+ e5√

2

4φ dZ2,

F(4) =

(

F(3) −1

g2h1 ∧ h2 ∧ h3 +

1√2 g

F i(2) ∧ hi

)

∧ dZ,

φ =1√2φ,

e−1 L6 = −1

4R +

1

2(∂φ)2 − 1

4e−

√2φ F i

µν Fi µν +

1

8g2e

√2φ +

1

12e−2

√2φ F 2

3 , (56)

where hi = σi − 1√2g Ai. The Lagrangian in Eq. (56) is precisely the N = 4g Romans’

theory.

We can study a similar problem to the one in the previous section. One can write

down a similar ansatz for the metric, i.e. a wrapped product of AdS4 × Σ2, where Σ2

is a an hyperbolic plane or a sphere, as before. The system of Eqs.(22)-(24) can be

studied in the massless case, i.e. m = 0, while g and a are non-vanishing quantities.

We have the solution

f = −ϕ, ϕ =g

4√

2r +

1

8log(r), h = − g

4√

2r +

3

8log(r) . (57)

Therefore, the six-dimensional metric is given by

ds2 =e− gr

2√

2

r1/4[dt2 − dr2 − dz2 − dv2 − rdΩ2

λ]. (58)

The metric has a singularity. One way to resolve it is to look for a solution that

has more degrees of freedom excited such as a non-Abelian solution. Another way to

have an acceptable gravity solution is to construct a black hole solution such that the

singularity is hidden by the horizon [21]. In the next section we will discuss a resolution

of the singularity in the S2 case, i.e., when the parameter λ = −1 in Eqs.(22) to (24).

15

4.1 A non-Abelian solution

We would like to present a non-Abelian solution that resolves the “bad” singularity of

the solution Eq.(58). This solution will be useful for the study of the gravity dual for

the three-dimensional N = 2 super Yang-Mills theory. Actually this solution, being

related to the solution in reference [9] represents a smeared NS-fivebrane on S2 after

a T-duality. In the IR the theory living on the brane will be N = 2 super Yang-

Mills theory in three dimensions plus Kaluza-Klein modes that do not decouple. The

solution in the string frame is

ds2 = 2 [dt2 − dr2 − d~x22 − e2 h(dθ2 + sin θ2dϕ2)],

F = −w′σ2dr ∧ dθ + w′ sin θσ1dr ∧ dϕ+ (w2 − 1) σ3 sin θdθ ∧ dϕ,

φS =1

2log

(

sinh(r)

R(r)

)

, w(r) =r

sinh(r), w′ =

dw

dr,

e2 h = R(r)2, R(r) =√

2r coth(r) − r2/ sinh(r)2 − 1 . (59)

This is the same solution reported by Chamseddine and Volkov in [32, 33].

The relations between the scalars and metric in Einstein frame and string frame are

ϕ = −1

2φS; gµν(E) = eφSgµν(S).

In Einstein frame the solution is

ds2E = 2eφS(−dt2 + dr2 + d~x2

2) + e2 G(r)(dθ2 + sin2 θdϕ2), (60)

where the dilaton is given by

φS =1

2log

(

sinh(r)

R(r)

)

,

R(r) =√

2 r coth(r) − w(r)2 − 1, e2 G = R(r)2, w(r) =r

sinh(r). (61)

The field strength components are

F 1(2) = w′ sin θdr ∧ dϕ,F 2

(2) = −w′dr ∧ dθ,F 3

(2) = (w2 − 1) sin θdθ ∧ dϕ . (62)

Collecting all we have the following ten-dimensional solution:

ds210 = e−

3√

2

8φ

[

−dt2 + dr2 + d~x22 + e2G(dθ2 + sin2 θdϕ2) +

1

g2

3∑

i=1

(σi − g√2Ai

(1))2

]

16

+e5√

2

8φdZ2

F(4) = (− 1

g2h1 ∧ h2 ∧ h3 +

1√2 g

F i(2) ∧ hi) ∧ dZ,

φ =1√2φ = −1

4log

(

sinh(r)

R(r)

)

. (63)

A black hole solution in the massless theory

The solutions of the massless six-dimensional Romans’ theory, the string frame met-

ric and 2-form ansatz are

ds2 =f(r)

rdt2 − 1

r f(r)dr2 − R2 (dθ2

1 + sin2 θ1 dϕ21) − dx2 − dy2,

F2 = h(r) dr ∧ dt+RQm√

2sin θ1 dθ1 ∧ dϕ1, (64)

where f and h are functions of r while γ and Qm are constants.

The solution in the string frame is

h(r) =Qe

r2, f(r) = −M +

2Q2e

r+

(

g2

2+Q2

m

R2

)

r, φ(r) =1

2log(r). (65)

Similar solutions were previously discussed in [34]. Our solution has a different geom-

etry and can carry both electric and magnetic charges.

One can up-lift the above solution to type IIA theory, and the results are

ds210 = −1

2r−1/8 ×

(

f(r)√rdt2 − 1√

r f(r)dr2 − R2

√r (dθ2

1 + sin2 θ1 dϕ21) −

√r (dx2 + dy2)

)

+r3/8

g2

(σ1)2 + (σ2)2 +

(

σ3 +g Qe√

2 rdt+

g RQm

2cos θ1 dϕ1

)2

+ r−5/8 dZ2,

φ = −1

4log(r),

F4 = − 1

g2σ2 ∧ σ2 ∧ h3 ∧ dZ +

1√2 g

F2 ∧ h3 ∧ dZ,

h3 = σ3 +g Qe√

2 rdt+

gRQm

2cos θ1 dϕ1 . (66)

For different values of the constants Qe, Qm, R, g we will have either a horizon or a

naked singularity. It should be instructive to compute the entropy and the Hawking

17

temperature of this black holes and comparing with the M-theory black holes obtained

in [35]. Since the system analyzed in that reference is the same in string variables as

the one we analyze here, it is expected that those results will be repeated. Indeed, the

ten-dimensional interpretation suggested in [35] agrees with the one we have described

above.

A solution with excited B-fields

Here we consider the six-dimensional Romans’ theory with the mass parameter and

all the fields except the scalar and the three-form field set to zero. In addition, we take

G3 to be a constant. The geometry of the AdS3 × R3 spacetime is given by

ds2 = e2 f (dt2 − dr2 − dz2) − e2 h (dx2 + dy2 + dv2) . (67)

Let us consider a spinor satisfying the following constraints

γ2γ7ǫi = ǫi γ456ǫi = ǫi , (68)

and

Gxyv = G . (69)

From the vanishing of the supersymmetric variation of gravitinos and gauginos we

obtain two independent equations

h′ = −ef

(

g

4√

2eϕ +

G

2e−3h+2ϕ

)

, (70)

f ′ = −ef

(

g

4√

2eϕ − G

2e−3h+2ϕ

)

, (71)

ϕ′ = −h′ . (72)

A fixed-point solution can be easily obtained

eφ√

2 =g e3h

2√

2G. (73)

The values of h(r) remain undetermined since the equations for φ′ and h′ are propor-

tional to each other. In this case, f(r) is given by

f(r) = − log

(

g2 e3h r

8G

)

, (74)

therefore the metric is given by

ds2 =1

(

g2 e3h r8 G

)2 (dt2 − dr2 − dz2) − e2h(dx2 + dy2 + dv2) . (75)

18

Equation (72) implies that ϕ = −h, where for simplicity we omit integration constants.

The system of equations above can be solved leading to

f(h) = −h+1

2log

(

g

4√

2e4h +

G

2

)

. (76)

Next, if we make the change of the integration variable r → h, such that

ef dr = − e5h

( g

4√

2e4h) + G

2

dh , (77)

in terms of h the metric reads

ds2 = − e10h

(

g

4√

2e4h + G

2

)2 dh2 + e−2h

(

g

4√

2e4h +

G

2

)

(dt2 − dz2)− e2h (dx2 + dy2 + dv2).

(78)

Using Eq.(16) we can write down the field strength of the dual theory as follows

Ftrz = −G e2√

2φ+3 f−3 h . (79)

The ten-dimensional solution is

ds210 = −1

2e√

2φ/4ds26 +

1

g2e−3

√2φ/4 (dθ2 + sin θ2dϕ2 + (dψ + cos θdϕ)2) + e5

√2φ/4 dz2

1 ,

F4 = −G e2√

2φ+3 f−3 h dt ∧ dr ∧ dz ∧ dz1 −1

g2sin θ dψ ∧ dθ ∧ dϕ ∧ dz1,

φ =1√2φ . (80)

The metric has a form of a warped product of a six-dimensional space, a three sphere

and a single coordinate. We also note that the G3 field does not play a similar role

to the previous cases. In fact, there is no twisting in this case, since the space is not

curved. One can view this solution as a NS-fivebrane of type IIB after wrapping one

of the directions on a circle. It would be interesting to find an example in which the

twisting on the curved manifold is performed by a B2 field.

5 Discussion

Here we would like to start with some comments about the solutions presented in

sections 3 and 4. The solutions in section 3 are very similar to those obtained in [10] for

supergravities in seven dimensions. This may not be a coincidence since in the string

19

frame both supergravities (one in seven dimensions and the other in six dimensions)

have very similar structure. Furthermore, their supersymmetric transformations also

look similar. This indicates that lower-dimensional supergravities are in some sense

related to each other. This relation may be better understood in the light of the up-

lifted solutions in ten or eleven dimensions since the up-lifted solutions could be related

by the known dualities.

In section 4 we constructed a non-Abelian solution that, as we noted, is identical

to the solution obtained in [32, 33] and interpreted in [9] as a wrapped NS-fivebrane.

In that case it was a gravity dual of a theory that is very similar to N = 1 SYM.

In our case, we can interpret the massless solution as the same NS-fivebrane with

one direction compactified. Therefore, the number of supercharges remains the same

and the world-volume theory is three-dimensional N = 2 SYM theory, as pointed out

before. The identity of the string frame action in six dimensions (with m = 0) and

the action in seven dimensions (with topological mass equal to zero) reveals that we

are dealing with the same system. We can extend this and try to look at the same

issue in five dimensions. In fact, a similar situation occurs there too. One of the

solutions presented in [36] can be related to the NS-fivebrane solution. In that paper,

another solution with electric fields is shown. It would be interesting to understand

this electromagnetic solution in the context of string theory.

We have analyzed the singularity structure of the equations representing D4-branes

(of a D4-D8 system) wrapping two- and three-cycles. Because it was not possible for

us to find exact solutions we turned to approximations inspired by ref. [10]. Thus, we

could find some solutions in Laurent series. We argued that they were non-acceptable as

the IR limit of a gauge theory. Expanding the solutions around r = 0 we have identified

the twisted theories on the brane-worldvolume and also found the operator inserted in

the UV of the gauge theory, deforming it and breaking some supersymmetries. It is

desirable to determine the precise form of the three (two) dimensional effective action

of the SCFT that results after wrapping the brane on the two- and three-cycles.

It would be very nice to find new solutions where more fields are excited. For example

solutions with the Abelian gauge field turned on could be interesting since it seems that

they will involve a non-trivial B2 field. It should also be useful to find solutions where

the twisting is realized by a Bp potential form (with p > 1). The relevance of this

type of solutions for non-commutative theories is evident. In this paper we found a

solution in section 4 that has only B2 fields excited. However, it does not represent

a twisted theory since the brane is not wrapped on a curved manifold. Moreover, its

ten-dimensional interpretation shows that we are dealing with a smeared NS-fivebrane.

The work of this paper together with the cases of M5, D3 and M2 branes constitute

the cases that complete the flows between AdS-like spacetimes in the UV and the

IR limit of gauge theories in lower dimensions. Finally, it would be interesting to

20

study systems with compactifications like the ones discussed here but in the context

of F (4) × G supergravity constructed in [37]. For that, it will be necessary to find a

prescription for up-lifting this supergravity to ten dimensions. The analysis of those

theories should give place to theories with enhanced ENf+1 symmetry, and it should

be very interesting to understand the relation between moduli spaces of Type I’ and

Heterotic strings from a gravitational point of view.

21

Acknowledgements

We would like to thank Freddy A. Cachazo, Jose Edelstein, Jerome P. Gauntlett,

Kentaro Hori, Andreas Karch, Nakwoo Kim, Ian I. Kogan, Juan Maldacena, Christo-

pher N. Pope, Arta Sadrzadeh, Le G.Q. Thong and Cumrun Vafa for helpful discus-

sions. The work of C.N. is supported by Fundacion Antorchas of Argentina. The

work of I.Y.P. is supported by the US Department of Energy under grant DE-FG03-

95ER40917. The work of M.S. is supported in part by funds provided by the U.S.

Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-

94ER40818, CONICET of Argentina, Fundacion Antorchas of Argentina and The

British Council. M.S. also acknowledges the kind hospitality of the Theoretical Physics

of University of Oxford where part of this work was carried out.

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